Rate of return
In finance, return is a profit on an investment. It comprises any change in value of the investment, and/or cash flows which the investor receives from that investment over a specified time period, such as interest payments, coupons, cash dividends and stock dividends. It may be measured either in absolute terms or as a percentage of the amount invested. The latter is also called the holding period return.
A loss instead of a profit is described as a negative return, assuming the amount invested is greater than zero.
To compare returns over time periods of different lengths on an equal basis, it is useful to convert each return into a return over a period of time of a standard length. The result of the conversion is called the rate of return.
Typically, the period of time is a year, in which case the rate of return is also called the annualized return, and the conversion process, described below, is called annualization.
The return on investment is return per dollar invested. It is a measure of investment performance, as opposed to size.
Calculation
The return, or the holding period return, can be calculated over a single period. The single period may last any length of time.The overall period may, however, instead be divided into contiguous subperiods. This means that there is more than one time period, each sub-period beginning at the point in time where the previous one ended. In such a case, where there are multiple contiguous subperiods, the return or the holding period return over the overall period can be calculated by combining the returns within each of the subperiods.
Single-period
Return
The direct method to calculate the return or the holding period return over a single period of any length of time is:where:
For example, if someone purchases 100 shares at a starting price of 10, the starting value is 100 x 10 = 1,000. If the shareholder then collects 0.50 per share in cash dividends, and the ending share price is 9.80, then at the end the shareholder has 100 x 0.50 = 50 in cash, plus 100 x 9.80 = 980 in shares, totalling a final value of 1,030. The change in value is 1,030 − 1,000 = 30, so the return is.
Negative initial value
Return measures the increase in size of an asset or liability or short position.A negative initial value usually occurs for a liability or short position. If the initial value is negative, and the final value is more negative, then the return will be positive. In such a case, the positive return represents a loss rather than a profit.
If the initial value is zero, then no return can be calculated.
Currency of measurement
The return, or rate of return, depends on the currency of measurement. For example, suppose a US$10,000 cash deposit earns 2% interest over a year, so its value at the end of the year is US$10,200 including interest. The return over the year is 2%, measured in USD.Let us suppose also that the exchange rate to Japanese yen at the start of the year is 120 yen per USD, and 132 yen per USD at the end of the year. The value in yen of one USD has increased by 10% over the period.
The deposit is worth 1.2 million yen at the start of the year, and 10,200 x 132 = 1,346,400 yen at the end of the year.
The return on the deposit over the year in yen terms is therefore:
This is the rate of return experienced either by an investor who starts with yen, converts to dollars, invests in the USD deposit, and converts the eventual proceeds back to yen; or for any investor, who wishes to measure the return in Japanese yen terms, for comparison purposes.
Annualization
Without any reinvestment, a return over a period of time corresponds to a rate of return :For example, let us suppose that US$20,000 is returned on an initial investment of US$100,000. This is a return of US$20,000 divided by US$100,000, which equals 20 percent. The US$20,000 is paid in 5 irregularly-timed installments of US$4,000, with no reinvestment, over a 5-year period, and with no information provided about the timing of the installments. The rate of return is 4,000 / 100,000 = 4% per year.
Assuming returns are reinvested however, due to the effect of compounding, the relationship between a rate of return, and a return over a length of time is:
which can be used to convert the return to a compound rate of return :
For example, a 33.1% return over 3 months is equivalent to a rate of:
per month with reinvestment.
Annualization is the process described above of converting a return to an annual rate of return, where the length of the period is measured in years and the rate of return is per year.
According to the CFA Institute's Global Investment Performance Standards,
This is because an annualized rate of return over a period of less than one year is statistically unlikely to be indicative of the annualized rate of return over the long run, where there is risk involved.
Annualizing a return over a period of less than one year might be interpreted as suggesting that the rest of the year is most likely to have the same rate of return, effectively projecting that rate of return over the whole year.
Note that this does not apply to interest rates or yields where there is no significant risk involved. It is common practice to quote an annualized rate of return for borrowing or lending money for periods shorter than a year, such as overnight interbank rates.
Logarithmic or continuously compounded return
The logarithmic return or continuously compounded return, also known as force of interest, is:and the logarithmic rate of return is:
or equivalently it is the solution to the equation:
where:
For example, if a stock is priced at US$3.570 per share at the close on one day, and at US$3.575 per share at the close the next day, then the logarithmic return is: ln = 0.0014, or 0.14%.
Annualization of logarithmic return
Under an assumption of reinvestment, the relationship between a logarithmic return and a logarithmic rate of return over a period of time of length is:so is the annualized logarithmic rate of return for a return, if is measured in years.
For example, if the logarithmic return of a security per trading day is 0.14%, assuming 250 trading days in a year, then the annualized logarithmic rate of return is 0.14%/ = 0.14% x 250 = 35%
Returns over multiple periods
When the return is calculated over a series of sub-periods of time, the return in each sub-period is based on the investment value at the beginning of the sub-period.Suppose the value of the investment at the beginning is, and at the end of the first period is. If there are no inflows or outflows during the period, the holding period return in the first period is:
If the gains and losses are reinvested, i.e. they are not withdrawn or paid out, then the value of the investment at the start of the second period is, i.e. the same as the value at the end of the first period.
If the value of the investment at the end of the second period is, the holding period return in the second period is:
Multiplying together the growth factors in each period and :
This method is called the time-weighted method, or geometric linking, or compounding together the holding period returns in the two successive subperiods.
Extending this method to periods, assuming returns are reinvested, if the returns over successive time subperiods are, then the cumulative return or overall return over the overall time period using the time-weighted method is the result of compounding all of the growth factors together:
If the returns are logarithmic returns, however, the logarithmic return over the overall time period is:
This formula applies with an assumption of reinvestment of returns and it means that successive logarithmic returns can be summed, i.e. that logarithmic returns are additive.
In cases where there are inflows and outflows, the formula applies by definition for time-weighted returns, but not in general for money-weighted returns.
Arithmetic average rate of return
The arithmetic average rate of return over time periods of equal length is defined as:This formula can be used on a sequence of logarithmic rates of return over equal successive periods.
This formula can also be used when there is no reinvestment of returns, any losses are made good by topping up the capital investment and all periods are of equal length.
Geometric average rate of return
If compounding is performed,, and if all periods are of equal length, then using the time-weighted method, the appropriate average rate of return is the geometric mean of returns, which, over n periods, is:The geometric average return is equivalent to the cumulative return over the whole n periods, converted into a rate of return per period. Where the individual sub-periods are each equal, and there is reinvestment of returns, the annualized cumulative return is the geometric average rate of return.
For example, assuming reinvestment, the cumulative return for four annual returns of 50%, -20%, 30%, and −40% is:
The geometric average return is:
The annualized cumulative return and geometric return are related thus: